Haboush's theorem (original) (raw)
In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that F(v) ≠ 0.
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| dbo:abstract | In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that F(v) ≠ 0. The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of V, and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p.When K has characteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of G implies that F can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic p was proved by W. J. , about a decade after the problem had been posed by David Mumford, in the introduction to the first edition of his book Geometric Invariant Theory. (en) Le théorème de Haboush est un théorème par lequel William Haboush a démontré une conjecture de Mumford, établissant que pour tout groupe algébrique réductif G sur un corps k, pour toute représentation de G sur un k-espace vectoriel V, et pour tout vecteur non nul v dans V fixe par l'action de G, il existe sur V un polynôme G-invariant F tel que F(v) ≠ 0 et F(0) = 0. Le polynôme peut être choisi homogène, c'est-à-dire élément d'une puissance symétrique du dual de V, et si la caractéristique de k est un nombre premier p > 0, le degré du polynôme peut être choisi égal à une puissance de p. Pour k de caractéristique nulle, ce résultat était bien connu : dans ce cas, (en) de Weyl sur la complète réductibilité des représentations de G garantit même que F peut être choisi linéaire. L'extension aux caractéristiques p > 0, conjecturée dans l'introduction de a été démontrée par . (fr) |
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| dbp:authorlink | Vladimir L. Popov (en) |
| dbp:first | V.L. (en) |
| dbp:id | M/m065570 (en) |
| dbp:last | Popov (en) |
| dbp:title | Mumford hypothesis (en) |
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| gold:hypernym | dbr:F |
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| rdfs:comment | In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that F(v) ≠ 0. (en) Le théorème de Haboush est un théorème par lequel William Haboush a démontré une conjecture de Mumford, établissant que pour tout groupe algébrique réductif G sur un corps k, pour toute représentation de G sur un k-espace vectoriel V, et pour tout vecteur non nul v dans V fixe par l'action de G, il existe sur V un polynôme G-invariant F tel que F(v) ≠ 0 et F(0) = 0. (fr) |
| rdfs:label | Haboush's theorem (en) Théorème de Haboush (fr) |
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